We analyse maps of the spatially-resolved nebular emission of $\approx$1500 star-forming galaxies at $z\approx0.6$-$2.2$ from deep KMOS and MUSE observations to measure the average shape of their rotation curves. We use these to test claims for declining rotation curves at large radii in galaxies at $z\approx1$-$2$ that have been interpreted as evidence for an absence of dark matter. We show that the shape of the average rotation curves, and the extent to which they decline beyond their peak velocities, depends upon the normalisation prescription used to construct the average curve. Normalising in size by the galaxy stellar disk-scale length ($R_{\rm{d}}$), we construct stacked position-velocity diagrams that trace the average galaxy rotation curve out to $6R_{\rm{d}}$ ($\approx$13 kpc, on average). Combining these curves with average HI rotation curves for local systems, we investigate how the shapes of galaxy rotation curves evolve over $\approx$10 Gyr. The average rotation curve for galaxies binned in stellar mass, stellar surface mass density and/or redshift is approximately flat, or continues to rise, out to at least $6R_{\rm{d}}$. We find a correlation between the outer slopes of galaxies' rotation curves and their stellar mass surface densities, with the higher surface density systems exhibiting flatter or less steeply rising rotation curves. Drawing comparisons with hydrodynamical simulations, we show that the average shapes of the rotation curves for our sample of massive, star-forming galaxies at $z\approx0$-$2.2$ are consistent with those expected from $\Lambda$CDM theory and imply dark matter fractions within $6R_{\rm{d}}$ of at least $\approx60$ percent.